The existence of a metastable minimum inside the photon-sphere implies the presence of a new class of quasi-bound states whose imaginary part, which is related to the tunnelling probability through the barrier, is in general very small. Since the imaginary part of the QNMs is related to the inverse of the life-time, we expect that this family of modes will dominate the late time of the ring-down phase (long-lived modes). As pioneered by Chandrasekhar and Ferrari [1]}
in the study of gravitational wave scattering on ultra-compact stars, quasi bound states will present as Breit-Wigner resonances in the scattering amplitude. More recently the resonance method has been used to compute long lived modes of AdS BHs [2]} and of ultra-compact configurations like constant density stars and gravastars [3]} whose effective potentials share astonishing analogies with the one for circular fuzz-ball, including the absence of a horizon. To implement these technique, our starting point is the choice of physically motivated boundary conditions.
The general asymptotic behaviour at infinity reads:
\(R(\rho )\sim B(\omega ) e^{-i \omega \rho }+C(\omega ) e^{i \omega \rho }\qquad \qquad \rho \rightarrow \infty \) 
